A hexagon is obtained by joining, in order, the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,1)$, $(2,0)$, and $(0,1)$. The perimeter of the hexagon can be written in the form $a+b\sqrt{2}+c\sqrt{5}$, where $a$, $b$ and $c$ are whole numbers.  Find $a+b+c$.
Solution: We must find the length of each side of the hexagon to find the perimeter.

We can see that the distance between each pair of points $(1, 2)$ and $(2, 2)$, $(2, 2)$ and $(2, 1)$, and $(2, 1)$ and $(3, 1)$ is 1. Thus, these three sides have a total length of 3.

We can see that the distance between $(0, 1)$ and $(1, 2)$ is $\sqrt 2$. The distance between $(3, 1)$ and $(2, 0)$ is also $\sqrt 2$. These two sides have a total length of $2\sqrt 2$.

We can see that the distance between $(2, 0)$ and $(0, 1)$ is $\sqrt 5$. Thus, the last side has length of $\sqrt 5$.

Summing all of these distances, we find that the perimeter is ${3 + 2\sqrt 2 + 1\sqrt 5}$, so $a+b+c=\boxed{6}$.